Asymptotic description of fast thermal processes in scalar harmonic lattices

Thermal processes in $d$-dimensional ($d$ = 1,2) scalar harmonic lattices with simple structure are considered. The redistribution between the averaged kinetic and potential energies of particles after instantaneous thermal excitation is described. At times much longer than the characteristic period of atomic oscillations, the difference between the kinetic and potential energies oscillates with amplitude decreasing by a power law (typically, the exponent is $-d$/2). The frequencies of the oscillations are determined from the dispersion relation for the lattice. The above results can be generalized for scalar and vector lattices of complex structure and various dimensions. As an example, the oscillations of the kinetic tmperature in the two-dimensional hexagonal lattice (graphene lattice) are investigated.